Grade 7 · Ratios & Proportional Reasoning · MAT-07-RP-001
Recognize Proportional Relationships
Represent and solve recognize proportional relationships problems using precise mathematical notation, models, and reasoning.
Learning objectives
What you should be able to do
- Represent and solve recognize proportional relationships problems using precise mathematical notation, models, and reasoning.
- Explain why a method for recognize proportional relationships works, interpret the result in context, and verify it independently.
Prerequisite check
Make sure the foundation is ready.
The new lesson depends on this prerequisite as active knowledge.
Learn
Build the idea from meaning, not memorization.
Mathematical meaning
Recognize Proportional Relationships is about multiplicative comparison. Equivalent representations preserve the same scale relationship, not merely the same difference.
Represent the relationship
The Math Box uses ratio table to expose the structure. Change one input, predict the result, then connect the visual change to an equation, table, graph, number line, or geometric model.
Calculate, interpret, and verify
Verify with an inverse operation, equivalent representation, estimate, or second method so both the calculation and reasoning are auditable.
Math Box · Interactive lesson
Touch the math. Change it. See what stays true.
Use the interactive model before and after the worked examples. Change the inputs, make a prediction, then use the model to test whether your reasoning holds.
Ratio Table Lab
Scale both quantities by the same factor and watch equivalent ratios stay aligned.
- Model one example of recognize proportional relationships in the Math Box.
- Change one input, predict the effect, and test the prediction.
- State the relationship or invariant that explains what stayed mathematically consistent.
Worked examples
Twenty different ways to see the concept work.
The pair (2, 4) follows y = 2x. Is the relationship proportional?
y/x = 4/2 = 2; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (3, 9) follows y = 3x. Is the relationship proportional?
y/x = 9/3 = 3; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (4, 16) follows y = 4x. Is the relationship proportional?
y/x = 16/4 = 4; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (5, 25) follows y = 5x. Is the relationship proportional?
y/x = 25/5 = 5; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (6, 36) follows y = 6x. Is the relationship proportional?
y/x = 36/6 = 6; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (7, 49) follows y = 7x. Is the relationship proportional?
y/x = 49/7 = 7; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (8, 64) follows y = 8x. Is the relationship proportional?
y/x = 64/8 = 8; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (9, 18) follows y = 2x. Is the relationship proportional?
y/x = 18/9 = 2; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (2, 6) follows y = 3x. Is the relationship proportional?
y/x = 6/2 = 3; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (3, 12) follows y = 4x. Is the relationship proportional?
y/x = 12/3 = 4; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (4, 20) follows y = 5x. Is the relationship proportional?
y/x = 20/4 = 5; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (5, 30) follows y = 6x. Is the relationship proportional?
y/x = 30/5 = 6; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (6, 42) follows y = 7x. Is the relationship proportional?
y/x = 42/6 = 7; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (7, 56) follows y = 8x. Is the relationship proportional?
y/x = 56/7 = 8; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (8, 16) follows y = 2x. Is the relationship proportional?
y/x = 16/8 = 2; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (9, 27) follows y = 3x. Is the relationship proportional?
y/x = 27/9 = 3; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (2, 8) follows y = 4x. Is the relationship proportional?
y/x = 8/2 = 4; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (3, 15) follows y = 5x. Is the relationship proportional?
y/x = 15/3 = 5; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (4, 24) follows y = 6x. Is the relationship proportional?
y/x = 24/4 = 6; yesA proportional relationship has a constant ratio y/x and includes the origin.
The pair (5, 35) follows y = 7x. Is the relationship proportional?
y/x = 35/5 = 7; yesA proportional relationship has a constant ratio y/x and includes the origin.
Practice
Guided practice with hints
01The pair (6, 48) follows y = 8x. Is the relationship proportional?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: y/x = 48/6 = 8; yes
02The pair (7, 14) follows y = 2x. Is the relationship proportional?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: y/x = 14/7 = 2; yes
03The pair (8, 24) follows y = 3x. Is the relationship proportional?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: y/x = 24/8 = 3; yes
04The pair (9, 36) follows y = 4x. Is the relationship proportional?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: y/x = 36/9 = 4; yes
05The pair (2, 10) follows y = 5x. Is the relationship proportional?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: y/x = 10/2 = 5; yes
06The pair (3, 18) follows y = 6x. Is the relationship proportional?
Hint: Represent the relationship first, keep units visible, and identify a second verification method.
Answer: y/x = 18/3 = 6; yes
Practice
Independent practice
01The pair (4, 28) follows y = 7x. Is the relationship proportional?
Answer: y/x = 28/4 = 7; yes
02The pair (5, 40) follows y = 8x. Is the relationship proportional?
Answer: y/x = 40/5 = 8; yes
03The pair (6, 12) follows y = 2x. Is the relationship proportional?
Answer: y/x = 12/6 = 2; yes
04The pair (7, 21) follows y = 3x. Is the relationship proportional?
Answer: y/x = 21/7 = 3; yes
05The pair (8, 32) follows y = 4x. Is the relationship proportional?
Answer: y/x = 32/8 = 4; yes
06The pair (9, 45) follows y = 5x. Is the relationship proportional?
Answer: y/x = 45/9 = 5; yes
07The pair (2, 12) follows y = 6x. Is the relationship proportional?
Answer: y/x = 12/2 = 6; yes
08The pair (3, 21) follows y = 7x. Is the relationship proportional?
Answer: y/x = 21/3 = 7; yes
Common mistakes
Learn to catch the error, not just the answer.
Check for a common scale factor or constant unit rate.
Convert both to a common unit rate first.
Applications & challenge
Use the idea beyond a single exercise.
Applications
- Compare prices, speeds, recipes, maps, taxes, discounts, tips, and rates using common units.
- Scale designs and predict related quantities while preserving multiplicative relationships.
Challenge problems
The pair (4, 32) follows y = 8x. Is the relationship proportional?
y/x = 32/4 = 8; yes
The pair (5, 10) follows y = 2x. Is the relationship proportional?
y/x = 10/5 = 2; yes
The pair (6, 18) follows y = 3x. Is the relationship proportional?
y/x = 18/6 = 3; yes
The pair (7, 28) follows y = 4x. Is the relationship proportional?
y/x = 28/7 = 4; yes
Flashcards
Retrieve it from memory.
Flashcards
Card 1 of 4
Game mode
Grade 7 Reasoning Sprint
Solve recognize proportional relationships problems accurately and justify each result with a valid check.
Practice
Mastery check
01The pair (8, 40) follows y = 5x. Is the relationship proportional?
Answer: y/x = 40/8 = 5; yes
02The pair (9, 54) follows y = 6x. Is the relationship proportional?
Answer: y/x = 54/9 = 6; yes
03The pair (2, 14) follows y = 7x. Is the relationship proportional?
Answer: y/x = 14/2 = 7; yes
04The pair (3, 24) follows y = 8x. Is the relationship proportional?
Answer: y/x = 24/3 = 8; yes
05The pair (4, 8) follows y = 2x. Is the relationship proportional?
Answer: y/x = 8/4 = 2; yes
06The pair (5, 15) follows y = 3x. Is the relationship proportional?
Answer: y/x = 15/5 = 3; yes
07The pair (6, 24) follows y = 4x. Is the relationship proportional?
Answer: y/x = 24/6 = 4; yes
08The pair (7, 35) follows y = 5x. Is the relationship proportional?
Answer: y/x = 35/7 = 5; yes
Terminology
Words to know
- ratio
- A multiplicative comparison between two quantities.
- unit rate
- A rate expressed per one unit of the second quantity.
- equivalent ratios
- Ratios made by scaling both quantities by the same nonzero factor.
- constant of proportionality
- The unit rate k in a proportional equation y = kx.
Continue learning
Connected concepts
Curriculum references
Standards and instructional references
California Mathematics Framework (2023)
Middle-school instructional guidance, mathematical practices, modeling, and coherent progression.Common Core State Standards for Mathematics — Grade 7
Grade-level expectations for ratios, number systems, algebra, geometry, statistics, and probability.Principles to Actions: Ensuring Mathematical Success for All
Research-informed emphasis on reasoning, representations, discourse, productive struggle, and conceptual understanding.